in radians, or α'=38F2/D
in arc minutes, for **D** in mm (α'=1.5F2/D
for **D** in inches). Evidently, unlike the linear quality field,
which only changes in proportion to **F**3,
quality angular field changes in proportion to **F**2
and in inverse proportion to the aperture diameter **D**.

As **Eq. 70** implies,
the wavefront error of coma in the Newtonian is inversely proportional
to the square of its ƒ-ratio for given angular field radius, and to the
third power of it for given linear field radius. Thus it decreases
exponentially with the increase in focal ratio, as illustrated on a
simplified scheme below.

**FIGURE 113**: The geometry of coma ray spot
formation in a fast (black) vs. slow mirror (blue) for given field
angle. The slower mirror has about twice smaller relative aperture,
thus its ray spot should be four times smaller (inversely proportional
to the square of ƒ-ratio
for given linear field radius),
but it is made larger for visual clarity.

Also, **Eq. 70** shows, coma of
the concave mirror is independent of its conic. It, however, changes with the
stop position
and object distance, as
described in 2.2. Coma (it also contains specifics on
the geometric, or ray aberration). In
terms of object distance **O** and mirror radius of curvature **R**,
the coma wavefront error changes in proportion to (O+R)/O, with **R**
being, according to the sign convention, negative. Thus, mirror coma
diminishes with the object distance, falling to zero for object at the
center of curvature (O=-R).

What is usually seen of
the coma in the eyepiece, with sufficiently bright stars, roughly
corresponds to the dense 1/3 of its geometric blur (sagittal coma). Its
angular size needs to be ~5 arc minutes for its shape to be recognized by an
average eye. Since, from Eq. 17,
(from Sa=S/ƒe)
angular sagittal coma in the eyepiece
(neglecting eyepiece distortion) is 215tanε/F2 arc minutes, with **ε**
being the apparent viewing field angle in the eyepiece, in degrees, the
eyepiece viewing
angle at which the geometric sagittal coma would become obvious to most people
is given by tanε~F2/43.
This means that one could see it at the edge of a 40° AFOV (w/AFOV=2ε) eyepiece at ƒ/4,
regardless of its focal length.

However, what we see
in the eyepiece is not a ray spot, but an actual diffraction image
deformed by the aberration. The above "geometric" angle at which
sagittal coma becomes visible is noticeably reduced in conventional eyepieces, due to
the significant additional blurring caused by
eyepiece astigmatism. Also, in order for the comatic
deformation to be seen in the actual star image, this image has to be
noticeably deformed by the aberration. For coma, that occurs as the
aberration decidedly exceeds ~1 wave P-V on the wavefront (or, as the
full size blur exceeds four times the Airy disc diameter, and the
sagittal coma is nearly 1.6 times the Airy disc diameter). At this
aberration level, and beyond, the actual visible blur of relatively
bright stars roughly resembles the form and size of geometric sagittal coma. As the error diminishes below ~1 wave P-V, the effect gradually
transforms into one-sided intensity distribution asymmetry visible in
the diffraction rings, but with the spurious disc
still well defined.

From **Eq. 70**, the field height at
which the coma error becomes large enough (~1.25 wave P-V) for the
central maxima deformation to begin to
resemble sagittal coma in the actual star image is about three times the
diffraction limit, or approximately
h~F3/30,
in mm. For an ƒ/4 mirror, that corresponds to ~2.1mm (**FIG.****
114**, top right).
Substituting h~F3/30 for **h** in the relation for angular sagittal
coma, Sa=S/ƒe,
gives Sa=F/480ƒe** **
in radians, or Sa=7.2F/ƒe
in arc minutes.

From Sa=5, the
eyepiece focal length needed to enlarge the actual comatic image of
a bright star with this amount of coma to 5 arc minutes apparent size,
is approximately ƒe~1.4F. Hence an average
observer, with a 5.6mm eyepiece and an ƒ/4 mirror, can expect to notice
coma on bright stars from ~2mm off-axis out. This corresponds to larger
than
20° field angles, as in the above consideration; the difference is that
that the implication here is that the shorter f.l. eyepieces would not
show edge coma due to the diminishing actual deformation of the central
maxima at the image height they reach.

With an
ƒ/6 mirror, it will
take an 8.4mm eyepiece to show the coma becoming apparent from some 7.2mm off-axis
(which, from ε~57.3h/ƒe,
** ƒ**e
being the ep f.l., would require 80+ degrees distortion-free AFOV);
higher magnification will reveal little of the characteristic comatic
deformation inside 7mm off-axis, due to the deformation becoming less
apparent with the decreasing wavefront error. Compromised image
sharpness, however, will be noticeable well within the 7mm-radius field
circle (**FIG 115**, bottom).

**
**

FIGURE
114: Ray spots and diffraction images of a 6"
ƒ/8.15 paraboloid and
a sphere (top) and the final ray spot after the eyepiece for a fast
and slow mirror (bottom).

(**A**).
For better resemblance to their real-life appearance, diffraction images are reduced by a factor of
3; also, they include 0.2D (linear) central obstruction effect. At 0.28° off-axis, the coma RMS wavefront error
of paraboloid - 0.075 wave (0.80 Strehl) is identical to the RMS
wavefront error of
spherical aberration of the sphere. The combined error of
the sphere at 0.28° degrees off-axis is 0.12 wave RMS (0.56
Strehl).

(**B**) Paraboloid's
coma is changing with its focal ratio **F**.
Linear
blur size varies inversely to **F**2,
while the Airy disc changes
in proportion to **F**, resulting in the wavefront error for
given field height to vary inversely to **F**3.
Thus, quality field size changes inversely to **F**2 angularly, and
inversely to **F**3
linearly.

(**C**)
Monochromatic (e-line) ray spots of a
paraboloid at ƒ/4.5 and
ƒ/9,
modified by the typical level of astigmatism and spherical aberration
induced by conventional eyepieces (for given RMS error, astigmatic
ray spot is 2.6 times
smaller than coma's). Eyepiece astigmatism changes inversely to the
telescope focal ratio** F** in the ray spot diameter, and
inversely to **F**2
as wavefront error.

Due to eyepiece astigmatism, which increases with the square of eyepiece field angle, comatic-like deformation
in the eyepiece grows rapidly toward the field edge. For given linear
field radius in the image plane, eyepiece
astigmatism increases in inverse proportion to the square of the focal ratio
as transverse aberration (i.e. ray spot size) and inversely to the cube
of it as wavefront error (lens astigmatism, given by
Eq. 22, increases with
the square of entrance beam width, which is inversely proportional to
mirror focal ratio, and inversely to lens' focal length, which is
proportional to the mirror focal ratio for given aperture and nominal
magnification). Therefore, its proportion to mirror coma at
given linear field radius remains constant regardless of mirror focal
ratio.

If we express the coma blur length (Eq.
17) in arc minutes, it comes to 645h/DF3.
Angular size of this blur on the retina is multiplied by telescope
magnification **M**, to 645hM/DF3.
Setting 645hM/DF3=3,
since it is the angular size at which an object begins to appear
non-point like to the average eye (although shape recognition requires
at least 5 arc minutes size),
gives the field radius **h** at which angular size of the coma blur
is three arc minutes as

h=DF3/215M.

Average P-V wavefront error of astigmatism in a conventional eyepiece is
approximated by WA~0.02ƒe(α/F)2
in units of 550nm wavelength, where **ƒ**e
is the ep focal length, and **
α** is the ep field
angle (AFOV/2). Substituting
α~57.3h/ƒe
gives WA~66h2/ƒeF2,
with the astigmatism wavefront error relating to that of coma (WC=37.9h/F3,
also in units of the wavelength) as 1.7hF/ƒe.

The combined visible blur size can be, at least
roughly, approximated as larger than coma-alone blur by a factor of 1+(WA/WC).
If we assume that only about the bright 1/3 of the coma blur (sagittal
coma) is visible, which is most often the case (the exception are bright
stars, especially in large apertures) then the combined blur is roughly
three times smaller.

Graph below (**FIG. 115**, top) illustrates how
the quality visual field in
the Newtonian telescope varies due to coma alone, but it also outlines
how the magnitude of astigmatism in conventional eyepieces affects the
actual field. Note that D/M in the
above equations is, for convenience, replaced by the pupil diameter **P**.
**FIG. 115** bottom illustrates quality of the field within which
comatic deformation of stellar images is not detectable, but the
aberration is still large enough to affect image contrast.

**FIGURE 115**:
TOP - Quality of the visual field in
Newtonian is mainly determined by two factors: mirror coma and eyepiece
astigmatism. Diffraction-limited field radius **h**L
for mirror coma (**blue plot**) is based on the wavefront error (WFE), but
says little about the actual star appearance. Whether coma will be
noticed, or not, depends primarily on the angular size of
the coma/astigmatism-bloated central diffraction maxima in the eyepiece.
Since it changes with the relative magnification of a telescope, it can
be expressed in terms of the eyepiece exit pupil diameter **P**. For
all but very bright telescopic stars it is the bright 1/3 tip of the
coma ray spot (sagittal coma) that approximates the visible aberration,
in which case the coma-free visual field is the largest (**green
plots**).
Assuming the entire coma spot (tangential coma) visible, the actual
visual diffraction-limited field radius (in a sense that within it there
won't be appreciable difference in the appearance of a star vs. aberration-free image), is smaller
than the nominal WFE-based field for exit pupils of about 2mm and
smaller, but larger for larger exit pupils (**red plots**). Assuming only sagittal portion of the coma ray spot visible, the actual visual
diffraction-limited field is larger than the nominal WFE-based field
already at around 1mm exit pupil diameter, and more so for larger exit
pupils. But, as mentioned, mirror coma is only one factor causing
off-axis bloating of star images. At larger relative apertures (~ƒ/5
and larger with conventional eyepieces) eyepiece astigmatism becomes
significant and/or dominant (w/conventional eyepieces). Average WFE of
eyepiece astigmatism (**W**A)
can be also roughly estimated, as well as its effect on the size of
combined coma/astigmatism blur. The inset with diffraction pattern illustrates how eyepiece
astigmatism combines with coma in producing the actual combined blur
(ray spot on white background corresponds to 1 wave P-V of coma, next at
right to it; the bottom 1/3 of the spot is sagittal coma). With an
average conventional eyepiece, the combined blur is approximated by
1+(1.7h/P) in units of the coma blur (either sagittal or tangential),
thus increasing exponentially with the field radius **h**. With
corrected eyepiece, the combined blur is roughly twice the comatic blur
alone. Neither is plotted, being only rough approximations, but it is
safe to conclude that the actual diffraction limited visual field for
mirror/eyepiece combinations is between the red and green plot for given
exit pupil: closer to the green for corrected eyepieces, and closer to the
red for the average conventional eyepiece.

BOTTOM - Graph above
describes the size of field in the Newtonian in terms of a noticeable
deformation of the stellar image. However, within a portion of the field
w/o noticeable deformation of star images, coma is still large enough to
affect performance. Starting at the field height where the coma deforms
the central maxima just enough to show the typical deformation under
sufficiently high magnification - F3/30
in mm, as mentioned before - will turn to MTF for the indication of
performance level toward field center (tangential MTF is for the blur
axis perpendicular to MTF bars, and sagittal MTF is for the blur and
bars in the same orientation; the average is approximately midway
between the two). It is not the level of aberration
present by itself, rather magnification used that determines when the
effect of aberration will become noticeable.

Magnification acts as a frequency filter, with increasing magnification
generally expanding frequency range toward higher frequencies,
ultimately reaching system's cutoff frequency. Assuming that the
threshold detection at cutoff frequency λ/D (in radians; 1.89/D in arc
minutes, for λ=0.00055mm and aperture diameter **D** in mm) requires
the separation between two consecutive bright lines to be 5 arc minutes,
1.89M/D=5, with **M** being the nominal telescope magnification;
taking relative magnification per mm of aperture m=M/D, it becomes
1.89m=5, and m=5/1.89=2.64 (67x per inch). Since the inverse of relative
magnification gives the corresponding eyepiece exit pupil diameter, the
"cutoff pupil" is about P=0.37mm. At h=F3/30
mm linear field radius (**red**),
angular size of the sagittal coma is, 215h/DF3
arc minutes which, after substituting for **h**, gives 7.2/D arc
minutes. The relative magnification m=M/D that will magnify it to 5 arc
minutes is obtained from 7.2m=5, thus m=0.7, and P=1.4. This
magnification is smaller than the system cutoff magnification by a
factor of 0.26, thus it is effectively a low-pass filter for frequencies
of ~0.26 and lower, with the higher frequencies attenuated. Still, as
the graph indicates, the contrast drop in this range is still quite
significant (since this field radius can be considered a median between
the inner field with no coma deformation detectable, and outer field
where it is obvious). The field graph top right shows that the linear
field radius with this much coma ranges from below 6mm at ƒ/4
to over 17mm at ƒ/8,
with the corresponding eyepiece apparent field radius needed to reach
this portion of mirror image from 22° to
nearly 70°, respectively, which means it
would be seen near edge of 50° AFOV eyepiece
(and more inside the field with wider-angle eyepieces) at ƒ/4,
but no eyepiece would reach that far out at ƒ/8
(recall that the ep f.l. needed to enlarge the coma to 5 arc minutes is
1.4F mm). For comparison, MTF graph show contrast drop at a three times
larger field radius (hence three times lower magnification, and
corresponding exit pupil P=4.2mm), where the cutoff frequency is below
0.09 (**black**). The magnitude of contrast drop toward the effective
cutoff frequency for the magnification is similar, but the resolution
limit is three times more coarse. Contrast is significantly lower
roughly in the higher half of the range of resolvable frequencies,
becoming similar to that at three times smaller field radius - and field
center - inside the lower half. Effective cutoff magnification is
achievable at the very edge of widest-field (~100°
AFOV) eyepieces at ƒ/4,
and somewhat farther from the edge with faster mirrors. At the field
radius twice smaller than the one at which the deformation of stellar
image due to coma becomes apparent, the effective cutoff is at 0.52
frequency, with the corresponding exit pupil P=0.7mm (**orange**).
There is no stellar deformation detectable this far off axis, but the
contrast drop is still significant. The radius is only about 1mm at ƒ/4,
increasing to 8.5mm at ƒ/8,
and the effect near cutoff frequency is seen at about 11°
apparent field radius at ƒ/4,
and 40° at ƒ/8.
At yet twice smaller field radius (F3/120
mm radius, **blue**), the entire range
of frequencies is covered, but contrast drop is less than the
diffraction-limited maximum (F3/90
mm radius).

The green plots on **FIG. 115** top, for the field radius with the
angular sagittal coma within 3 arc minutes, represent the field
quality with zero eyepiece astigmatism. The red plots, for the entire
coma blur (tangential coma) are probably more representative with
respect to the actual blur size, at least
as a rough guide, considering that eyepiece astigmatism commonly
enlarges the final blur. Somewhat larger blur, 3-5 arc minutes is still
acceptable, and toward field edge, as much as 6-9 arc minutes is
tolerable.

Expectedly, apparent size of the comatic image is in
inverse proportion to the exit pupil size, i.e. relative telescope
magnification. But it doesn't translate into inferior field performance.
For instance, at the exit pupil diameter P=1 and ƒ/5,
coma blur reaches 3 arc minutes at nearly 0.6mm off axis. However, the
corresponding error is still more than twice smaller than "diffraction
limited". Since 1mm exit pupil at ƒ/5
requires 5mm f.l. eyepiece (i.e. f.l.=F), its field radius at a standard
~45° AFOV is ~2.1mm, implying that roughly
the inner half of the field area (~3/4 of the field radius) will be
better than diffraction limited. At the very edge, coma is only about
1/3 larger than diffraction limited, or about 0.55 wave P-V, with the
angular size of about 10 arc minutes.

On the other hand, at three times lower magnification
(P=3), the 3-minute coma blur field radius also triples, to 1.8mm, with
the coma this far off now somewhat worse than diffraction limited. At
the field edge (assuming the same ~45° AFOV),
it is as much as 1.9 wave P-V, or 10 arc minutes, same as at 1mm exit
pupil. The difference is that both, comatic pattern deformation and
energy lost to the rings are significantly greater at 1.9 than at 0.55
wave P-V. In other words, coma alone is more noticeable and detrimental
at the lower magnification.

Plugging in the eyepiece astigmatism factor, with the
astigmatism P-V wavefront error approximately **W**A~1.7hWC/P,
where **h** is the field radius and **W**C
the P-V wavefront error of coma, makes the actual blur larger. At 1mm
exit pupil (P=1), edge coma at h=2.1mm is about 0.55 wave P-V, implying
about 2 waves P-V of eyepiece astigmatism; at 3mm exit pupil (P=3),
h=6.2 and **W**C=2,
implying as much as 7 waves of astigmatism. In both cases eyepiece
astigmatism dominates mirror coma by roughly the same ratio, about 3.5
(eyepiece astigmatism for given field angle increases with the focal
length, and coma increases with the image field radius), but both coma
and astigmatism are significantly larger at lower magnification. Using

Bcomb=[1+(WA/WC)]BC

approximation for the combined geometric blur, with **B**C
being the coma blur, gives that the combined edge blur for for both, 1mm
and 3mm exit pupil is over four times greater than that for coma alone.
That comes to over 40 arc-minute blur, more than the naked-eye full
Moon. Of course, not all of it is visible, even with bright stars;
roughly, only about 1/3 of it, or less, can be seen. That is still
nearly half the full Moon diameter; since in both cases the eyepiece
astigmatism is strong, and by nearly the same ratio so than coma, degree
of deformation of the actual pattern is similar in both cases,
resembling the right most pattern above, with somewhat flatter base and
a fainter triangular tail.

The difference is that the blur size relative to the
Airy disc is three times smaller at 1mm exit pupil, since at 1mm exit
pupil everything is three times more magnified than at 3mm. So, while
the apparent size of stellar images is similar to that at 3mm exit
pupil, energy concentration vs. image scale at the higher magnification
is significantly better, and the bloated outer-field stars are roughly
three times smaller relative to near-perfect mid-field stars.

The 1 wave P-V wavefront error of coma is characterized by Sidgwick as
limit to tolerable, visually. Obviously, it may not be so at high enough
magnifications, and/or with significant eyepiece astigmatism present. In
a 200mm ƒ/5 Newtonian, coma is that strong at 3.3mm off axis. In the
10mm f.l. eyepiece, tangential coma is already 8.5 arc minutes, and
sagittal near to 3. Average eyepiece astigmatism P-V wavefront error at
this field height (close to 20° field angle) is nearly three times
stronger than coma's, enlarging sagittal coma to roughly 11 arc minutes
- quite obvious for most observers.

Summing it up, stellar deformation due to coma is not
apparent at any magnification within field radius less than three times
the diffraction-limited coma field. The field radius at which coma
becomes apparent, if sufficiently magnified, is given by F3/30
in mm, or tanα=F2/39
angularly, with the magnification needed to make it apparent to the eye
(~5 arc minutes sagittal coma) being provided by an eyepiece of 1.4F mm
focal length. This assumes negligible eyepiece astigmatism. With
conventional eyepieces, stellar deformation in faster mirrors will
progressively more deteriorate toward the outer field due to eyepiece
astigmatism. For instance, linear field radius at which stellar
deformation becomes apparent is more than three times larger at ƒ/6
than ƒ/4, but the corresponding
angular radius in the eyepiece magnifying it to 5 arc minutes is only
about twice larger at ƒ/6 (**FIG.
115** top right). Thus the astigmatism at ƒ/6
is effectively somewhat more than doubled at twice the angular radius
(quadrupled due to doubled apparent angle, and larger by a factor 1.5
due to larger e.p. f.l. but more than halved due to the slower focal
ratio), while at ƒ/4 it is
quadrupled, and combined with twice larger coma.

●
**lower-order astigmatism**, from
Eq. 18,
after substitutions (for θ=0), as the P-V and RMS wavefront error at
the best focus, for
object at infinity, are: